Face hallucination via patch-pairs leaning based methods has been wildly used in the past several years.Some position-patch based face hallucination methods have been proposed to improve the representation power of image patch and obtain the optimal regressive weighted vector.The rationale behind the position-patch based face hallucination is the fact that human face is always highly structured and consequently positioned and it plays an increasingly important role in the reconstruction.However,in the existing position-patch based methods,the probe image patch is usually represented as a linear combination of the corresponding patches of some training images,and the reconstruction residual is usually measured using the vector norm such as 1-norm and 2-norm.Since the vector norms neglect two-dimensional structures inside the residual,the final reconstruction performance is not very satisfactory.To cope with this problem,we present a weighted nuclear-norm constrained sparse coding (WNCSC) model for position-patch based face hallucination.In addition,an efficient algorithm for the WNCSC is developed using the altating direction method of multipliers (ADMM) and the method of augmented Lagrange multipliers (ALM).The advantages of the proposed model are twofold:in order to fully make use of low-rank structure information of the reconstruction residual,the weighted nuclear norm is applied to measure the residual matrix,which is able to alleviate the bias between input patches and training data,and it is more robust than the Euclidean distance (2-norm);the more flexible selection method for rank components can determine the optimal combination weights and adaptively choose the relevant and nearest hallucinated neighbors.Finally,experimental results prove that the proposed method outperforms the related state-of-the-art methods in both quantitative and visual comparisons.